PHYS 1220, Engineering Physics, Chapter 30 – Inductance
Instructor: TeYu Chien
Department of Physics and Astronomy
University of Wyoming
Goal of this chapter is to learn how the magnetic
inductance behaves in circuits
- Mutual Inductance between two coils nearby.
•
ϵ 2=−N 2
d Φ B2
;
dt
ϵ1 =−N 1
d Φ B1
dt
(Faraday's law of induction)
• We know that the magnetic flux Φ B1 and Φ B2 is
proportional to the current of the other coil. (Note:
for Φ B2 , Φ B2=∮ B⃗1⋅d ⃗A , where B1 is the
magnetic field produced by coil #1 on the axis of coil
#1 as: B1=
μ 0 N 1 I 1 a2
2 (x 2+a 2)3 /2
(Eq, 28.16 on page 933).
Though this equation is only valid on the axis, we
could know that the magnetic field is proportional to
the current.)
• We can write the relationship between Φ B2 and
I 1 as: N 2 Φ B2=M 21 I 1 (and also for
N 1 Φ B1=M 12 I 2 ). And it could be prove that
M 21=M 12 , and we call it mutual inductance, M. So does the relationship:
M=
N 2 Φ B2 N 1 Φ B1
=
I1
I2
d I1
;
dt
• The first two equations could be rewritten as: ϵ2=−M
ϵ1 =−M
d I2
dt
• Unit for the mutual inductance: henry (H)
1 H =1Wb / A=1V⋅s / A=1 Ω⋅s=1 J / A
2
• The mutual inductance is the physics how the transformer works.
- Self-inductance and Inductors
• The magnetic field produced by a current in a circuit, if it changes over time, it
could induce emf (Ampere's Law). This is the origin of the self-induction, L.
•
•
N ΦB
I
d ΦB
dI
ϵ=−N
=−L
dt
dt
L=
(The first equal notation is Ampere's Law. This equation
describes the self induced emf)
• In other words, when the current-carrying wire is bended into coil shape and
placed in a circuit, the time-dependent current will produce a self-induced
emf and affect the circuit behavior.
• The notation used for inductor in circuit is:
dI
• This self-induced emf in circuit is described as: Δ V =L
dt
You Do Example 30.3 on page 997.
I Do Exercise 30.15 (a) on page 1014.
- Magnetic field energy stored in inductor
dU
di
=P=Vi= Li
•
dt
dt
I
•
dU = Li di ; so
1 2
U =∫ Li di = LI
2
0
2
• energy density (energy per volume) stored in magnetic field: u= U = B
V
2 μ0
- Circuits: (R-L circuit, L-C circuit, and R-L-C circuit)
• R-L circuit
•
I ( t)= ϵ (1−e−(R / L)t ) (When S1 closed
R
while S2 opened). Current increases and
saturates .
•
I ( t)=I 0 e−(R / L)t (When S2 closed while
opened) Current decays exponentially.
L
• Time constant: τ=
R
S1
•
•
•
L-C circuit
q (t)=Q cos (ω t+ϕ)
I ( t)=−ω Q sin(ω t+ϕ)
• oscillation frequency, f, and phase frequency, ω : ω=2 π f =
√
1
LC
• The current and charge are oscillating back and forth between capacitor and
inductor. The energy is transferred back and forth between electric field (in
capacitor) and magnetic field (in inductor).
• R-L-C circuit
• We know that the resistor, R, is the
damping source (consume the energy);
while capacitor and inductor store energy
in them. When combine these three
together, you should expect to see the
oscillation of current and charge between
capacitor and inductor with damping
through the resistor.
•
q (t)= A e−(R /2L )t cos(
√
1
R2
− 2 t+ϕ)
LC 4L
• oscillation phase frequency in R-L-C
circuit: ω ' =
√
1
R2
− 2
LC 4L
Math Preview for Chapter 31:
• Trigonometric functions
• Oscillation as functions of time
Questions to think about for Chapter 31:
• We only discussed the circuits with dc voltage (current). And we know that the
induction depends on the change of current as function of time. What if the ac
voltage (sin function) is applied to the circuit. How will the circuit react?